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Theta Nullvalues of Supersingular Abelian varieties

Andreas Pieper

Abstract

Let $η$ be a polarization with connected kernel on a superspecial abelian variety $E^g$. We give a sufficient criterion which allows the computation of the theta nullvalues of any quotient of $E^g$ by a maximal isotropic subgroup scheme of $\ker(η)$ effectively. This criterion is satisfied in many situations studied by Li and Oort. We used our method to implement an algorithm that constructs supersingular curves of genus 3.

Theta Nullvalues of Supersingular Abelian varieties

Abstract

Let be a polarization with connected kernel on a superspecial abelian variety . We give a sufficient criterion which allows the computation of the theta nullvalues of any quotient of by a maximal isotropic subgroup scheme of effectively. This criterion is satisfied in many situations studied by Li and Oort. We used our method to implement an algorithm that constructs supersingular curves of genus 3.
Paper Structure (26 sections, 38 theorems, 103 equations)

This paper contains 26 sections, 38 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Theta Groups
  3. Symmetric line bundles
  4. Theta structures
  5. Representations of a theta group
  6. Theta Nullvalues of principally polarized Abelian Varieties
  7. Complex analytic comparison theorem
  8. Finite commutative group schemes
  9. Truncated Witt group schemes
  10. Duality of truncated Witt group schemes
  11. Dieudonné theory
  12. Representation theory of finite commutative group schemes
  13. Completely decomposed $\Theta_{\eta}$-divisors
  14. Definition and geometric interpretation
  15. Rational equivalence of completely decomposed $\Theta_{\eta}$-divisors
  16. ...and 11 more sections

Key Result

Lemma 2.3

$K(\mathcal{L}^n)=[n]^{-1}(K(\mathcal{L}))$

Theorems & Definitions (113)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8
  • ...and 103 more